The gerund phrase, "seeing her," belongs to Annabell. Since Annabell is used as an possessive adjective modifying the gerund phrase, an apostrophe is necessary.
See Rule 12 in this site.
http://www.grammarbook.com/punctuation/apostro.asp
I was reading this sentence:
She was moving in and out without Annabell's seeing her.
Why does Annabell's have an apostrophe s?
3 answers
The word "seeing" is a gerund in this sentence; a gerund is a verb form being used as a noun. Annabell's is in possessive form in order to modify "seeing." If you don't have the possessive form in front of the gerund, the meaning is that "she was moving in and out without Annabell" ("Annabell" being the object of the preposition "without") -- and that's not what the sentence means. The object of the preposition "without" is "seeing" not "Annabell."
http://grammar.ccc.commnet.edu/grammar/gerunds.htm#possessive